I'll bet it would stick. You could sell it to schools by offering a bit of the profits to assist the athletic departments and offset use of state funds to prop up ADs. But again, private schools might be at a disadvantage.
I see sunday play again being an issue for us - if games were held at LES. A saturday only league might fly. What about some other nearby site - like Rio Tinto?
I would bet that the church would shy away from allowing naming/logo rights to a pro team that plays even once in a while on sunday - along with all the other unsavory things that are part of pro sports (alcohol sales, scantily-clad cheerleaders, perception of for-profit $ grab...). Might affect the church's non-profit status if they were associated.
Better to trademark "Team Brigham", have similar color scheme, etc and do it without naming/logo rights, but with the understanding that its cougar alumni/byu family.
I also think it would be wise to open up eligibility to any alumnus (graduated or not), or family of alums/faculty/staff. Players from small schools without teams could "earn eligibility" by enrolling part time. Might help players who never got their degree...

But i love the general concept. I would definitely watch, and buy gear of a team like that.